Given a group $G$ and a subgroup $H\subseteq G$, let $\overline{H}=\langle ghg^{-1}\mid h\in H\text{ and } g\in G\rangle$ be the normal closure of $H$ and $N(H)=\{g\in G\mid gHg^{-1}=H\}$ be the normalizer of $H$. What is the relationship between quotient groups $G/\overline{H}$ and $N(H)/H$?
It is trivial for an abelian group such that $G/\overline{H}=N(H)/H$ as $\overline{H}=H$ and $N(H)=G$. Does this equality can also be extended to non-abelian group? Or there is a simple counter example for this (better to be a finite group)? If they are not equal, do they have any other relationships?